Tangent Calculator (tan)

Find the tangent of an angle using our tool. Start by entering the angle and selecting whether it is in degrees or radians.

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What is the Tangent Function?

In trigonometry, the tangent function, abbreviated as tan, is one of the six fundamental trigonometric functions. For an acute angle in a right-angled triangle, the tangent is defined as the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle. This relationship is a cornerstone of trigonometry and is often remembered by the mnemonic “TOA,” which stands for Tangent is Opposite over Adjacent.

Beyond right triangles, the tangent function can be defined for any angle using the unit circle. If you draw an angle in standard position on a coordinate plane, its terminal side intersects the unit circle at a point (x, y). The tangent of that angle is the ratio y/x, which also represents the slope of the terminal side. This broader definition allows us to find the tangent of any angle, including those greater than 90° or less than 0°.

The Tangent Formula and Explanation

The primary formula for tangent in a right triangle is straightforward.

tan(θ) = Opposite / Adjacent

It can also be expressed using sine and cosine, which is useful for calculations involving the unit circle.

tan(θ) = sin(θ) / cos(θ)

The variables in these formulas are defined as follows:

Variable Explanations

Variable	Meaning	Unit	Typical Range
θ (theta)	The angle of interest.	Degrees or Radians	-∞ to +∞
Opposite	The length of the side across from angle θ in a right triangle.	Length (e.g., cm, inches)	Positive values
Adjacent	The length of the non-hypotenuse side next to angle θ.	Length (e.g., cm, inches)	Positive values

Practical Examples

Example 1: Finding the Height of a Tree

Imagine you are standing 50 meters away from the base of a tree. You look up at the top of the tree, and the angle of elevation from the ground to the top is 30°. How tall is the tree?

• Inputs: Angle (θ) = 30°, Adjacent side = 50m
• Formula: tan(30°) = Opposite / 50m
• Calculation: Opposite = 50m * tan(30°). Using a calculator, tan(30°) ≈ 0.577.
• Result: Opposite = 50 * 0.577 ≈ 28.85 meters. The tree is approximately 28.85 meters tall. A right triangle calculator can solve these problems instantly.

Example 2: Using Radians

An engineer is designing a ramp. The specifications require the ramp to have an angle of π/6 radians. What is the tangent of this angle?

• Inputs: Angle (θ) = π/6 radians
• Units: The input is already in radians. You may need to convert degrees to radians first if the angle was given in degrees.
• Calculation: tan(π/6). Since π/6 radians is equal to 30°, the result is the same as the previous example.
• Result: tan(π/6) ≈ 0.577.

How to Use This Tangent Calculator

Our calculator makes finding the tangent of an angle simple and quick. Follow these steps for an accurate result:

1. Enter the Angle: Type the numerical value of the angle into the “Angle Value” field.
2. Select the Unit: Use the dropdown menu to choose whether your angle is in “Degrees (°)” or “Radians (rad)”. This is a critical step, as the calculation changes depending on the unit.
3. View the Result: The calculator automatically computes and displays the tangent value. No need to press a “calculate” button.
4. Interpret the Output: The main result is shown in large text. Below it, you’ll find the angle converted to the other unit (e.g., if you entered degrees, it will show the radian equivalent).
5. Reset or Copy: Use the “Reset” button to clear the input and return to the default value. Use the “Copy Results” button to save the output to your clipboard.

Key Factors That Affect Tangent

Several factors influence the value of the tangent function:

• Angle Unit: The most common error is using the wrong unit. tan(45°) = 1, but tan(45 rad) ≈ 1.62. Always ensure your calculator is in the correct mode (Degrees or Radians).
• Asymptotes: The tangent function is undefined at certain angles, specifically at 90° (π/2 rad), 270° (3π/2 rad), and any angle that is a multiple of 180° (π rad) away from these points. At these values, the function has a vertical asymptote.
• Periodicity: The tangent function is periodic with a period of 180° (or π radians). This means tan(θ) = tan(θ + 180°). For example, tan(20°) is the same as tan(200°).
• Quadrant of the Angle: The sign (positive or negative) of the tangent value depends on the quadrant in which the angle’s terminal side lies. It’s positive in Quadrants I and III, and negative in Quadrants II and IV.
• Relationship to Sine and Cosine: Since tan(θ) = sin(θ) / cos(θ), the tangent is directly affected by the values of sine and cosine. When cos(θ) is zero, the tangent is undefined. See our sine calculator and cosine calculator for more.
• Inverse Function: The inverse tangent function, arctan or tan⁻¹, is used to find an angle when you know the tangent value. However, its output is restricted to the range of -90° to +90° (-π/2 to +π/2).

Frequently Asked Questions (FAQ)

1. What is tan(90°)?

The tangent of 90 degrees is undefined. This is because tan(90°) = sin(90°) / cos(90°) = 1 / 0, and division by zero is not possible. On the graph, this appears as a vertical asymptote.

2. Can the tangent of an angle be greater than 1?

Yes. Unlike sine and cosine, whose values are restricted to [-1, 1], the tangent value can be any real number, from negative infinity to positive infinity.

3. How do I find the tangent without a calculator?

For common angles like 0°, 30°, 45°, 60°, and 90°, you can use the properties of special right triangles (30-60-90 and 45-45-90) to find the ratio of the opposite to adjacent sides. For other angles, a calculator or a Pythagorean theorem calculator is necessary.

4. Why is my calculator giving the wrong answer for tangent?

The most likely reason is that your calculator is in the wrong mode. If your angle is in degrees, make sure your calculator is set to “DEG” and not “RAD”.

5. What’s the difference between tangent and arctangent?

Tangent (tan) takes an angle and gives a ratio. Arctangent (arctan or tan⁻¹) takes a ratio (the tangent value) and gives back an angle.

6. What is tangent used for in real life?

Tangent is used extensively in architecture, engineering, physics, and computer graphics. It helps calculate heights of objects, slopes of lines, angles of elevation, and trajectories.

7. Is tan(x) the same as the tangent line in calculus?

No. The trigonometric function tan(x) is a ratio of sides in a triangle. A tangent line in calculus is a straight line that “just touches” a curve at a single point, representing the instantaneous rate of change (derivative) at that point.

8. What is the value of tan(0)?

tan(0) = 0. This is because sin(0) = 0 and cos(0) = 1, so tan(0) = 0 / 1 = 0.